ATANH

Compute the inverse hyperbolic tangent, defined on the open interval (-1, 1).

Category: mathReturns: DOUBLEDialect: Standard

Syntax

ATANH(expr)

Description

## Overview Returns the inverse hyperbolic tangent of the input. ATANH is defined on the open interval (-1, 1), asymptotes to +/-infinity at the endpoints, and is equivalent to 0.5 * LN((1 + x) / (1 - x)). It is the inverse of TANH across its entire range. ATANH is central to special relativity (rapidity is the inverse hyperbolic tangent of velocity as a fraction of c), to Fisher's z-transform in statistics (stabilizing variance of correlation coefficients), and to logit-like transforms that map probabilities in (0, 1) to the real line. ## Behavior - Domain: (-1, 1). At +/-1 the result is +/-infinity. Outside the interval the result is NULL. - Range: all real numbers. - Returns NULL if the argument is NULL. - Is anti-symmetric: ATANH(-x) = -ATANH(x). - Is strictly increasing and continuous on its open domain. ## Numeric precision - Uses the IEEE 754 double-precision atanh routine, accurate to within one ULP over most of the domain. - Near the endpoints the derivative diverges, so small input noise produces large output swings. - The naive closed form 0.5 * LN((1 + x) / (1 - x)) loses precision near x = +/-1 because of cancellation; use ATANH directly. ## Compatibility - Conforms to the SQL standard definition of ATANH as a scalar DOUBLE function with domain (-1, 1). - Matches typical mathematical library implementations.

Parameters

Name	Type	Description
`expr`		Specifies the real-valued input in the open interval (-1, 1). At +/-1 the result is +/-infinity; outside the interval the result is NULL.

Examples

-- Basic literal
SELECT ATANH(0);
-- Result: 0.0

-- Inverse hyperbolic tangent of 0.5
SELECT ATANH(0.5);
-- Result: 0.5493061443340549

-- Round trip: ATANH(TANH(x)) = x for small x
SELECT ATANH(TANH(1.0));
-- Result: 1.0

-- Out-of-domain input returns NULL
SELECT ATANH(2);
-- Result: NULL

-- Closed form: ATANH(x) = 0.5 * LN((1 + x) / (1 - x))
SELECT ATANH(0.5), 0.5 * LN((1 + 0.5) / (1 - 0.5));
-- Result: 0.5493061443340549, 0.5493061443340549

-- Column use: recover rapidity from velocity (as fraction of c)
SELECT track_id, ATANH(velocity_over_c) AS rapidity
FROM physics.simulation.tracks
WHERE ABS(velocity_over_c) < 1;

Pitfalls

Inputs at exactly +/-1 return +/-infinity, which can poison downstream aggregates. Clamp with GREATEST(LEAST(x, 1 - eps), -1 + eps) when rounding noise can drive the input to the boundary.
Inputs outside [-1, 1] silently return NULL. Always validate upstream.
ATANH(TANH(x)) can return infinity for large x because TANH saturates to 1.0 in floating-point. Round-trip fidelity requires preserving the original x.
The naive closed form loses precision near x = +/-1. Prefer the built-in ATANH to the expansion.